150 lines
4.4 KiB
C++
150 lines
4.4 KiB
C++
/*
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* This Source Code Form is subject to the terms of the Mozilla Public
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* License, v. 2.0. If a copy of the MPL was not distributed with this
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* file, You can obtain one at http://mozilla.org/MPL/2.0/.
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*
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* Copyright 2015 Danny Robson <danny@nerdcruft.net>
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*/
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#include "polynomial.hpp"
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#include "maths.hpp"
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#include <limits>
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#include <cmath>
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static const size_t NEWTON_ITERATIONS = 1u;
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//-----------------------------------------------------------------------------
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namespace cruft::polynomial {
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template <>
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std::array<float,1>
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roots (std::array<float,2> coeff)
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{
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const float a = coeff[0];
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const float b = coeff[1];
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if (almost_zero (a))
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return { std::numeric_limits<float>::quiet_NaN () };
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return { -b / a };
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}
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}
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//-----------------------------------------------------------------------------
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namespace cruft::polynomial {
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template <>
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std::array<float,2>
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roots (std::array<float,3> coeff)
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{
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const float a = coeff[0];
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const float b = coeff[1];
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const float c = coeff[2];
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if (almost_zero (a)) {
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auto s = roots<1> ({b, c});
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return { s[0], std::numeric_limits<float>::quiet_NaN () };
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}
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auto descriminator = std::sqrt (pow (b,2) - 4 * a * c);
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return {
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(-b - descriminator) / (2 * a),
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(-b + descriminator) / (2 * a)
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};
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}
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}
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//-----------------------------------------------------------------------------
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// From graphics gems: http://tog.acm.org/resources/GraphicsGems/gemsiv/vec_mat/ray/solver.c
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namespace cruft::polynomial {
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template <>
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std::array<float,3>
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roots (std::array<float,4> coeffs)
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{
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const float _a = coeffs[0];
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const float _b = coeffs[1];
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const float _c = coeffs[2];
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const float _d = coeffs[3];
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// Take care of degenerate quadratic cases. We can also pass off if 'd'
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// is zero, but the benefit isn't clear given we have to merge results
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// at the end anyway.
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if (almost_zero (_a)) {
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auto s = roots<2> ({_b, _c, _d});
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return {s[0], s[1], std::numeric_limits<float>::quiet_NaN () };
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}
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std::array<float,3> s;
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// Normalise to x^3 + ax^2 + bx + c = 0
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const float a = _b / _a;
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const float b = _c / _a;
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const float c = _d / _a;
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// Substituate x = y - a / 3 to eliminate the quadric. Now: x^3 + px + q = 0
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const float p = (-a * a / 3.f + b) / 3.f;
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const float q = (2 * a * a * a / 27.f - a * b /3.f + c) / 2.f;
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// Polynomial descriminant
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const float D = q * q + p * p * p;
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// Solve using Cardano's method
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if (almost_zero (D))
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{
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if (almost_zero (q)) {
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s[0] = 0.f;
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s[1] = std::numeric_limits<float>::quiet_NaN ();
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s[2] = std::numeric_limits<float>::quiet_NaN ();
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} else {
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const float u = std::cbrt (-q);
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s[0] = 2 * u;
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s[1] = -u;
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s[2] = std::numeric_limits<float>::quiet_NaN ();
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}
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} else if (D < 0) {
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const float phi = std::acos (-q / std::sqrt (-p * p * p)) / 3.f;
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const float t = 2 * std::sqrt (-p);
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s[0] = t * std::cos (phi);
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s[1] = -t * std::cos (phi + pi<float> / 3.f);
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s[2] = -t * std::cos (phi - pi<float> / 3.f);
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} else {
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float u = std::cbrt (std::sqrt (D) + abs (q));
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if (q > 0.f)
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s[0] = -u + p / u;
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else
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s[0] = u - p / u;
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s[1] = std::numeric_limits<float>::quiet_NaN ();
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s[2] = std::numeric_limits<float>::quiet_NaN ();
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}
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// Resubstitute a / 3 from above
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const float sub = a / 3.f;
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for (auto &i: s)
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i -= sub;
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// Run some iterations of Newtons method to make the results slightly
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// more accurate, they're a little loose straight out of the bat.
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const float da = 3 * _a;
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const float db = 2 * _b;
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const float dc = 1 * _c;
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for (auto &i: s) {
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for (size_t j = 0; j < NEWTON_ITERATIONS; ++j) {
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float deriv = da * i * i + db * i + dc;
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if (almost_zero (deriv))
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continue;
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i = i - eval (coeffs, i) / deriv;
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}
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}
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return s;
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}
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}
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